Hmm. Your comment has brought to my attention an issue I hadn’t thought of before.
Are you familiar with Aumann’s knowledge operators)? In brief, he posits an all-encompassing set of world states that describe your state of mind as well as everything else. Events are subsets of world states, and the knowledge operator K transforms an event E into another event K(E): “I know that E”. Note that the operator’s output is of the same type as its input—a subset of the all-encompassing universe of discourse—and so it’s natural to try iterating the operator, obtaining K(K(E)) and so on.
Which brings me to my question. Let E be the event “you are a thing that thinks”, or “you exist”. You have read Descartes and know how to logically deduce E. My question is, do you also know that K(E)? K(K(E))? These are stronger statements than E—smaller subsets of the universe of discourse—so they could help you learn more about the external world. The first few iterations imply that you have functioning memory and reason, at the very least. Or maybe you could take the other horn of the dilemma: admit that you know E but deny knowing that you know it. That would be pretty awesome!
Or maybe you could take the other horn of the dilemma: claim that you know E but you don’t know that you know it. That would be pretty awesome!
When I was younger, a group of my friends started teasing others because they didn’t know the Hindu-Arabic number system. In reality, of course, they did know it, but they didn’t know that they knew it—that was the joke.
My question is, do you also know that K(E)? K(K(E))?
I have a sensory/gut experience of being a thinking being, or, as you put it, E.
Based on that experience, I develop the abstract belief that I exist, i.e., K(E).
By induction, if K(E) is reliable, then so is K(K(K(K(K(K(K(E)))))))). In other words, there is no particular reason to doubt that my self-reflective abstract propositional knowledge is correct, short of doubting the original proposition.
So I like the distinction between E and K(E), but I’m not sure what insights further recursion is supposed to provide.
I wasn’t familiar with this description of “world states”, but it sounds interesting, yes. I take it that positing “I am a think that things” is the same as asserting K(E). In asserting K(K(E)), I assert that I know that I know that I am a thing that thinks. If this understanding is incorrect, my following logic doesn’t apply.
I would argue that K(K(E)) is actually a necessary condition for K(E). Because if I don’t know that I know proposition A, then I don’t know proposition A.
Edit/Revised: I think all you have to do is realize that “K(K(A)) false” permits “K(A) false”. At first I had a little proof but now it seems just redundant so I deleted it.
So I guess I disagree, I think the iterations K(K...) are actually weaker statements, which are necessary for K(A) to be achieved. Consequentially I don’t see how you can learn anything beyond K(A).
K(A) is always a stronger statement than A because if you know K(A) you necessarily know A. (To get the terms clear: a “strong” statement corresponds to a smaller set of world states than a “weak” one.) It is debatable whether K(K(A)) is always equivalent to K(A) for human beings. I need to think about it more.
Format definition of K(E)={s \in S | P(s) \subset E }, where P is partition of S, ensures that K(K(E))=K(E). It’s easy to see: if s \in K(E) then P(s) \subset e, thus s \in K(K(E)), and similarly for s \notin K(E).
As for informal sence, I don’t see much use of K(K(E)) where E is a plain fact, if I aware that I know E, introspecting on that awareness will provide as much K-s as I like and little more. If I don’t aware that I know E (deep buried memory?), I will be aware of it when I remember it. But If I know that I know some class of facts or rules, that is useful for planning. However I can’t come up with useful example for K(K(K())) and higher.
Addition: Aumann’s formalization have limitations: it can’t represent false knowledge, memory glitches (when I know that I know something, but I can’t remember it), meta-knowledge, knowledge of rules of any kind (I’m not completely sure about rules).
Hmm. Your comment has brought to my attention an issue I hadn’t thought of before.
Are you familiar with Aumann’s knowledge operators)? In brief, he posits an all-encompassing set of world states that describe your state of mind as well as everything else. Events are subsets of world states, and the knowledge operator K transforms an event E into another event K(E): “I know that E”. Note that the operator’s output is of the same type as its input—a subset of the all-encompassing universe of discourse—and so it’s natural to try iterating the operator, obtaining K(K(E)) and so on.
Which brings me to my question. Let E be the event “you are a thing that thinks”, or “you exist”. You have read Descartes and know how to logically deduce E. My question is, do you also know that K(E)? K(K(E))? These are stronger statements than E—smaller subsets of the universe of discourse—so they could help you learn more about the external world. The first few iterations imply that you have functioning memory and reason, at the very least. Or maybe you could take the other horn of the dilemma: admit that you know E but deny knowing that you know it. That would be pretty awesome!
When I was younger, a group of my friends started teasing others because they didn’t know the Hindu-Arabic number system. In reality, of course, they did know it, but they didn’t know that they knew it—that was the joke.
I have a sensory/gut experience of being a thinking being, or, as you put it, E.
Based on that experience, I develop the abstract belief that I exist, i.e., K(E).
By induction, if K(E) is reliable, then so is K(K(K(K(K(K(K(E)))))))). In other words, there is no particular reason to doubt that my self-reflective abstract propositional knowledge is correct, short of doubting the original proposition.
So I like the distinction between E and K(E), but I’m not sure what insights further recursion is supposed to provide.
I just saw this and realized I basically just expanded on this above.
I wasn’t familiar with this description of “world states”, but it sounds interesting, yes. I take it that positing “I am a think that things” is the same as asserting K(E). In asserting K(K(E)), I assert that I know that I know that I am a thing that thinks. If this understanding is incorrect, my following logic doesn’t apply.
I would argue that K(K(E)) is actually a necessary condition for K(E). Because if I don’t know that I know proposition A, then I don’t know proposition A.
Edit/Revised: I think all you have to do is realize that “K(K(A)) false” permits “K(A) false”. At first I had a little proof but now it seems just redundant so I deleted it.
So I guess I disagree, I think the iterations K(K...) are actually weaker statements, which are necessary for K(A) to be achieved. Consequentially I don’t see how you can learn anything beyond K(A).
K(A) is always a stronger statement than A because if you know K(A) you necessarily know A. (To get the terms clear: a “strong” statement corresponds to a smaller set of world states than a “weak” one.) It is debatable whether K(K(A)) is always equivalent to K(A) for human beings. I need to think about it more.
Format definition of K(E)={s \in S | P(s) \subset E }, where P is partition of S, ensures that K(K(E))=K(E). It’s easy to see: if s \in K(E) then P(s) \subset e, thus s \in K(K(E)), and similarly for s \notin K(E).
As for informal sence, I don’t see much use of K(K(E)) where E is a plain fact, if I aware that I know E, introspecting on that awareness will provide as much K-s as I like and little more. If I don’t aware that I know E (deep buried memory?), I will be aware of it when I remember it. But If I know that I know some class of facts or rules, that is useful for planning. However I can’t come up with useful example for K(K(K())) and higher.
Addition: Aumann’s formalization have limitations: it can’t represent false knowledge, memory glitches (when I know that I know something, but I can’t remember it), meta-knowledge, knowledge of rules of any kind (I’m not completely sure about rules).